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We are given two points with coordinates \((q, 2 - q)\) and \((2q + 2, q - 4)\), and the line joining these points is perpendicular to the line \(3x - 4y + 5 = 0\).

To find the value of \(q\), let's go through the following steps:

### Step 1: Find the slope of the given line \(3x - 4y + 5 = 0\).
We need to rewrite this equation in slope-intercept form \(y = mx + b\), where \(m\) is the slope:

\[
3x - 4y + 5 = 0 \quad \Rightarrow \quad -4y = -3x - 5 \quad \Rightarrow \quad y = \frac{3}{4}x + \frac{5}{4}
\]

So, the slope of the given line is \(\frac{3}{4}\).

### Step 2: Find the slope of the line joining the points \((q, 2 - q)\) and \((2q + 2, q - 4)\).
The formula for the slope of a line between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substitute the coordinates of the points:

\[
\text{slope} = \frac{(q - 4) - (2 - q)}{(2q + 2) - q} = \frac{q - 4 - 2 + q}{2q + 2 - q}
\]
\[
\text{slope} = \frac{2q - 6}{q + 2}
\]

### Step 3: Use the perpendicularity condition.
If two lines are perpendicular, the product of their slopes is \(-1\). Let the slope of the given line be \(\frac{3}{4}\) and the slope of the line joining the points be \(\frac{2q - 6}{q + 2}\). Set up the equation:

\[
\frac{3}{4} \times \frac{2q - 6}{q + 2} = -1
\]

### Step 4: Solve for \(q\).
Now, solve the equation:

\[
\frac{3(2q - 6)}{4(q + 2)} = -1
\]

Multiply both sides by \(4(q + 2)\) to eliminate the denominator:

\[
3(2q - 6) = -4(q + 2)
\]

Expand both sides:

\[
6q - 18 = -4q - 8
\]

Move all terms involving \(q\) to one side and constants to the other side:

\[
6q + 4q = 18 - 8
\]
\[
10q = 10
\]

Solve for \(q\):

\[
q = 1
\]

### Final Answer:
The value of \(q\) is \(1\), which corresponds to option **C**.

Would you like further details on this process or additional explanations?

Here are five related questions you might want to explore:
1. How do you find the equation of a line perpendicular to a given one?
2. What is the general condition for two lines to be perpendicular?
3. How do you calculate the midpoint between two points in coordinate geometry?
4. Can you find the equation of a line passing through two points?
5. What happens when two lines are parallel?

**Tip:** When working with slopes, always remember that perpendicular slopes are negative reciprocals of each other!

The line joining the points with coordinates (q, 2 - q) and (2q + 2, q - 4) is perpendicular to the line with equation 3x - 4y + 5 = 0. What is the value of q?

This problem involves finding the value of q where the line joining points (q, 2 - q) and (2q + 2, q - 4) is perpendicular to the line 3x - 4y + 5 = 0. Explore key concepts in coordinate geometry, including slope calculations and the perpendicular slope condition. Suitable for Grades 10-12.

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